The home side skittled England for 143 before reaching 136-3 by stumps in their second innings in Hamilton, enjoying easily their best day of a series in which they had been outplayed until now.
Kane Williamson was at the crease on 50 and Rachin Ravindra two.
Williamson put on 89 for the second wicket with Will Young (60) following the departure of captain Tom Latham for 19.
Scoreless nightwatchman Will O'Rourke gave England captain Ben Stokes (2-45) a second wicket soon before the close.
New Zealand's controlled strokeplay in the final session was in contrast to chaotic English batting between lunch and tea where they lost their last eight wickets for 66 runs.
Pace bowler O'Rourke (3-33) ripped out three key wickets in a ferocious spell after lunch, before Matt Henry (4-48) and Mitchell Santner (3-7) instigated a late collapse.
Jacob Bethell (12) was the first to fall to O'Rourke, caught at point, before the towering quick prised out the ICC ranked first and second-best Test batters, Harry Brook and Joe Root.
After scoring match-changing centuries in each of the two big England wins, in Christchurch and Wellington, Brook lasted only one ball.
He played on to an in-swinger, prompting a roar of approval from the small crowd on a hot, still day at Seddon Park.
Root departed for 32 in O'Rourke's next over, caught at point off a limp cut shot from a delivery that bounced sharply.
Santner reinforced New Zealand's commanding position by removing Ollie Pope (24) and Stokes (27) in his first seven deliveries after the pair had put on 52 for the sixth wicket.
Pope edged to Daryl Mitchell at first slip before Stokes departed lbw, losing balance as he attempted a sweep shot.
The last five wickets fell for nine runs but, ahead by 204 runs, New Zealand declined the opportunity to enforce the follow-on.
Earlier, England reached 54-2 at lunch.
Zak Crawley fell to Henry for the fifth time in five innings this series, caught-and-bowled for 21 to continue a miserable run of form.
Crawley's score was his highest of the series, having managed only managed 47 runs at an average of 9.6.
The impressive Henry struck again in the same over, trapping fellow opener Ben Duckett lbw for 11.
England had taken more than an hour to claim the last New Zealand wicket after the home side resumed their first innings at 315-9.
New Zealand faced a further 15.1 overs and scored 32 runs before top-scorer Santner was bowled for 76 by Matthew Potts (4-90), having added 26 to his overnight score.
